how to display custom taxonomies in front page

I’d love to help you out. You’ll probably want to use the get_terms() function and it may look something like this:

$artists = get_terms(array(
    'taxonomy' => 'artist', // Taxonomy name, or array of taxonomy names
    'hide_empty' => false, // Show terms not assigned to anything
    'orderby' => 'name', // Order the artists by the term name
    'order' => 'DESC' // Order them alphabetically from A to Z
));

You can read more about what parameters are accepted here. That code snippet will pull all artists, and then on the page, you can display them in a table like this:

if ( is_wp_error($artists) ) {

    // Do something about the error

} elseif( count($artists) ) {

    // Artists found, open the table
    printf(
        '<table><thead><tr><th colspan="2">%s</th></tr><tr><th>%s</th><th>%s</th></tr></thead><tbody>',
        __('Artists'), 
        __('Name'), 
        __('Count')
    );

    // Loop through each artist, adding their row
    foreach( $artists as $artist ) {
        printf(
            '<tr><td>%s</td><td>%s<td></tr>',
            esc_html($artist->name),
            esc_html($artist->count)
        );
    }

    // Close the table
    echo('</tbody></table>');
} else {

   // No artists found

}

That example will display a table with two columns, one with the artist’s name and then the total number of music post they’re assigned to. Since each artist in the loop is a WP_Term object, you can access these properties:

object(WP_Term) (11) {
    ["term_id"]=>  //int
    ["name"]=>   //string 
    ["slug"]=>  //string 
    ["term_group"]=>  //int
    ["term_taxonomy_id"]=> //int
    ["taxonomy"]=>   //string
    ["description"]=>    //string
    ["parent"]=> //int
    ["count"]=>  // int
    ["filter"]= //string
    ["meta"]= array(0) {} //an array of meta fields.

I personally love modularity, so if you want to save writing repetitive code, you could do something like this:

// Define all the custom taxonomies you want to get and their nice label
$taxonomies = array(
   'albums' => __('Albums'),
   'artists' => __('Artists')
);

// Loop through them
foreach($taxonomies as $taxonomy => $taxonomy_name) {

    // Get the terms
    $terms = get_terms(array(
        'taxonomy' => $taxonomy, // Taxonomy name, or array of taxonomy names
        'hide_empty' => false, // Show terms not assigned to anything
        'orderby' => 'name', // Order by the term name
        'order' => 'DESC' // Order them alphabetically from A to Z
    ));
    if ( is_wp_error($terms) ) {

        // Do something about the error

    } elseif( count($terms) ) {

        // Terms found, open the table
        printf(
            '<table><thead><tr><th colspan="2">%s</th></tr><tr><th>%s</th><th>%s</th></tr></thead><tbody>',
            $taxonomy_name, // The nice label
            __('Name'), 
            __('Count')
        );

        // Loop through each term, adding their row
        foreach( $terms as $term ) {
            printf(
                '<tr><td>%s</td><td>%s<td></tr>',
                esc_html($term->name),
                esc_html($term->count)
            );
        }

        // Close the table
        echo('</tbody></table>');
    } else {

       // No terms found
       printf(
            __('No results found for <strong>%s</strong>.'), 
            $taxonomy_name
        );

    }
}